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7f37411 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import OAI.MeasureTheory.Falconer.Shells.HighShellParentEnergy
import OAI.MeasureTheory.Falconer.Shells.HighShellProfileMass
import OAI.MeasureTheory.Falconer.Shells.HighShellMeasurable
import OAI.MeasureTheory.Falconer.Shells.HighShellErrorPower
import OAI.MeasureTheory.Falconer.Shells.ShellApproximationError
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory Filter
namespace PositiveDistanceData
theorem uniform_high_shell_parent_bound {sig lam eps T : ℝ}
(hsig : 0 < sig) (heps : 0 < eps) :
∃ n N₀ : ℕ, ∀ d : PositiveDistanceData, ∀ (B : ℕ) (δ : ℝ),
RegularizedProfileData d.Q d.ρ d.N B δ d.lam d.mass →
N₀ ≤ d.N → d.sig = sig → d.lam = lam →
0 < d.Q.side → d.StronglySeparated → 0 < d.q →
d.sig*d.N ≤ (d.q:ℝ) → d.a+100*d.q < d.t → d.t ≤ d.N →
d.X.length = d.a → d.Y.length = d.a →
∀ v : ℝ, (d.t:ℝ)-d.sig*d.N ≤ v → v ≤ d.t →
∀ Ψ : ℝ → ℝ, Measurable Ψ → (∀ x, 0 ≤ Ψ x) →
(∀ x, Ψ x ≠ 0 → x ∈ Ioo (1/4:ℝ) 4) →
∀ W : ℝ, 0 ≤ W → (∀ x, Ψ x ≤ W) →
(∀ x ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ x) →
∀ H : ℝ, 0 ≤ H →
(∀ j ∈ Finset.range (n+1),
∀ b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList), ∀ ξ,
(d.inversePhasedHF v Ψ b.2 ξ).toReal ≤ H) →
(2:ℝ)^(-(d.a:ℝ)) *
(∫ r in Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)), ‖charFun d.weightedDistance r‖^2) ≤
((2:ℝ)^d.N)^eps*H+((2:ℝ)^d.N)^(-T) := by
obtain ⟨n,N₁,h₁⟩ := uniform_high_shell_profile_error (lam := lam) (T := (T+3)/2) hsig
obtain ⟨N₂,h₂⟩ := uniform_spatialLeibniz_sum_subpower 96 2004 (by norm_num) (by norm_num)
(show 0 < eps/4 by linarith) n
obtain ⟨N₃,h₃⟩ := fixed_constant_absorbed (half_pos heps) 4
refine ⟨n,max 1 (max N₁ (max N₂ N₃)),?_⟩
intro d B δ hd hN hsig' hlam hQ hsep hq hqs hw ht hX hY v hv hvt Ψ hΨ hpos hs W hW hΨW hone H hH0 hH
let := hd.probability
let := d.weightedDistance_isFinite hQ hsep
have hN1 : N₁ ≤ d.N := (le_max_left _ _).trans ((le_max_right _ _).trans hN)
have hN2 : N₂ ≤ d.N := (le_max_left _ _).trans ((le_max_right _ _).trans ((le_max_right _ _).trans hN))
have hN3 : N₃ ≤ d.N := (le_max_right _ _).trans ((le_max_right _ _).trans ((le_max_right _ _).trans hN))
let R : ℝ := (2:ℝ)^d.N
let μ := volume.restrict (Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)))
let e := R^(-((T+3)/2))
have herr : ∀ᵐ r ∂μ, ‖charFun d.weightedDistance r-d.normalizedHighShellF n v r‖ ≤ e := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with r hr
exact h₁ d B δ hd hN1 hsig' hlam hQ hsep hq hqs hw ht hX hY v r hv hr.1
have hi : Integrable (fun r => ‖charFun d.weightedDistance r‖^2) μ :=
integrable_norm_square_bounded μ _ measurable_charFun measureReal_nonneg
(Eventually.of_forall (fun r => norm_charFun_le r))
have hsq := (shell_square_approximation μ _ _ (d.normalizedHighShellF_measurable hQ hsep n v)
hi (show 0 ≤ e by dsimp [e]; positivity) herr).2
have hmain := d.normalizedHighShellF_parent_energy hQ hsep (by rw [hsig']; exact hsig.le)
hq hqs hw hv n Ψ hΨ hpos hs hW hΨW hone hH0 hH
let S := ∑ j ∈ Finset.range (n+1), spatialLeibnizCost 96 2004 (by norm_num) (by norm_num) j
(fun k => d.mass k-(k:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t
have hS : S ≤ R^(eps/4) := h₂ d.N hN2 _ d.q d.a d.t hq
have hS0 : 0 ≤ S := by
apply Finset.sum_nonneg
intro j hj
apply mul_nonneg (spatialFourierCost_nonneg ..)
apply mul_nonneg _ (by positivity)
apply List.sum_nonneg
intro x hx
obtain ⟨b,hb,rfl⟩ := List.mem_map.mp hx
positivity
have hcost : 4*S^2 ≤ R^eps := by
calc
_ ≤ R^(eps/2)*(R^(eps/4))^2 :=
mul_le_mul (h₃ d.N hN3) (pow_le_pow_left₀ hS0 hS 2) (sq_nonneg _) (by positivity)
_ = _ := by
rw [← Real.rpow_mul_natCast (show 0 ≤ R by dsimp [R]; positivity),
← Real.rpow_add (show 0 < R by dsimp [R]; positivity)]
congr 1
ring
have herror := high_shell_error_power d.N d.a ((le_max_left _ _).trans hN)
(hvt.trans (Nat.cast_le.mpr ht)) T
calc
_ ≤ (2:ℝ)^(-(d.a:ℝ)) * (2*(∫ r, ‖d.normalizedHighShellF n v r‖^2 ∂μ)+2*e^2*μ.real univ) :=
mul_le_mul_of_nonneg_left hsq (by positivity)
_ = 2*((2:ℝ)^(-(d.a:ℝ))*(∫ r, ‖d.normalizedHighShellF n v r‖^2 ∂μ))+
(2:ℝ)^(-(d.a:ℝ))*(2*e^2*μ.real univ) := by ring
_ ≤ 2*(2*S^2*H)+R^(-T) :=
add_le_add (mul_le_mul_of_nonneg_left hmain (by norm_num)) herror
_ ≤ R^eps*H+R^(-T) := by nlinarith [mul_le_mul_of_nonneg_right hcost hH0]
end PositiveDistanceData
end PlanarFalconer
end OAI
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